Reprinted from the 40th INTERNATIONAL WIRE AND CABLE SYMPOSIUM 1991.
OPTICAL FIBER FAILURE PROBABILITY PREDICTIONS
FROM LONG-LENGTH STRENGTH DISTRIBUTIONS
G. S. Glaesemann
Corning Incorporated
Corning New York
Abstract
Strength data from a recently developed apparatus for
measuring long-length fiber strength distributions are
analyzed in terms of proof test theory for truncated
distributions. Data are fitted using Weibull statistics and
scaled for bending and tensile lengths ranging from 1 meter
to 100 kilometers. Most tensile applications require strength
data near the proof stress level. For failure probability levels
less than lx10-5 most bending applications need be
concerned with flaws near the proof stress level.
Introduction
A schematic of a 1986 strength distribution of 16.5
kilometers of titania-doped silica-clad fiber proof tested to 50
kpsi (345 MPa) is shown in Figure 1.1 Observe that after
testing nearly 17 kilometers of fiber the distribution lacks
flaws that are of the greatest reliability risk for most
applications namely those near the proof stress level. For
the purposes of reliability prediction near the proof stress
level much more fiber must be tested. Also the amount of
time required to manually create multi-kilometer strength
distributions using common industry test methods makes the
creation of such distributions costly.
The purpose of this paper is to present a recently developed
technique for measuring the strength distribution near the
proof stress level and to examine how one might use these
data for making reliability predictions.
Creating a Long-Length Strength
A complete description of the operation of the equipment is
given in reference 2 and only a brief description will be given
here. The test apparatus is shown schematically in Figure 3
and consists of a proof testing machine for paying out fiber
into the gauge length under low loads. The gauge length
consists of 20 meters of fiber which starts at point A on the
payout tractor travels around a remote pulley assembly and
back to point B on the take-up tractor. The pulley assembly
consists of a pulley mounted on a load cell both of which are
attached to a pneumatic slide. Fiber is payed out under low
load into the gauge length after which the pulley assembly
moves on the slide and the fiber is loaded to a predetermined
maximum load level. As soon as the maximum load is
reached the pulley returns to its original position. The load
is carefully monitored during the entire load pulse and if
failure occurs the breaking load is recorded. Typical load
pulses are shown in Figure 4 for fiber that passes and fails the
test. If the fiber passes the load pulse test another 20 meter
length is indexed into the gauge length and the load pulse is
repeated. The loading and unloading rates are in the 200 to
400 kpsi/s (1400 to 2800 MPa/s) range and therefore the
probability of subcritical crack growth during testing is high.
Using the above apparatus 386 kilometers of titania-doped
silica-clad fiber4,5 proofed to 50 kpsi (350 MPa) were tested
to a maximum stress level of 350 kpsi (2450 MPa) in
approximately 4 weeks. All testing was carried out under
ambient conditions (20°C, 60% RH). The number of
recorded failures below 350 kpsi (2450 MPa) was 106 out of
a total 19,300 individual 20 meter tests. The failure
probability, F, was assigned to each fiber failure using the
median rank method,
F=
(I-0.3)
(J+0.4)
(1)
Distribution
A new fiber strength testing method was recently developed
for obtaining data on many kilometers of fiber in a more
timely fashion.2 It is believed that this test method will
enable engineers to better assess the failure probability of
flaws near the proof stress level. As shown in Figure 1, 95%
of the flaws on 20 meter gauge lengths have strengths greater
than 500 kpsi (3500 MPa) and therefore do not present a
reliability risk for long-length applications. The approach
taken in the development of a long-length strength
distribution was to avoid testing the strong flaws to failure.
This was accomplished by loading fibers during tensile
testing to a maximum load below the high strength region as
shown schematically in Figure 2, where the maximum load
during testing is set to break all flaws below 400 kpsi (2800
MPa) and pass those that are stronger.
where I is the fiber rank ranging from 1 for the weakest to
106 for the strongest fiber, and J is the total number of tests;
namely, 19,300. The data are shown in Figure 5 as a Weibull
plot of lnln(l/1-F) versus lnσf, where σf is the fracture
strength. The upper end of the strength distribution stops at
the maximum stress level of 350 kpsi, as planned, and the
lower end extends to a stress level slightly above the proof
stress of 50 kpsi (350 MPa). The above data demonstrate the
capability of obtaining multi-kilometer strength distributions
in a relatively short period of time.
The theoretical shape for inert strength distributions of proof
tested specimens is shown in Figure 6 from reference 3. The
pre-proof distribution has a constant slope m. Flaws away
International Wire & Cable Symposium Proceedings 1991 819
from the proof stress are unaffected by the proof stress and
are shown to follow the pre-proof distribution. Those that
grow subcritically during proofing have a post-proof slope of
n-2, where n is the well known fatigue susceptibility
parameter from the power law crack velocity model. Finally
the distribution is truncated near the proof stress level as
indicated by a vertical line. For fast unloading rates the
truncation strength is no less than 90% of the proof stress.3
The distribution in Figure 5 differs from the theoretical
distribution in that our testing demonstrates subcritical crack
growth due to a fatigue environment. In the case where
fatigue occurs during strength testing, the truncation strength
level will be less than the proof stress, simply because flaws
that just pass the proof stress will grow during subsequent
strength testing and fail at a stress level below the proof
stress. The flaws in the (n-2) slope region will, after crack
growth during the strength test, end up with a slope n+1.6
The data in Figure 5 are linear above the 125 kpsi (875 MPa)
stress level with a slope, m, of approximately 2. Below 125
kpsi (875 MPa), the data indicate the transition to a higher
slope region prior to truncation shown in Figure 6. However,
too few flaws were obtained in this region to accurately
determine where a slope of n+1 begins, obviously, many more
kilometers of data are needed.
Tensile Loading.
For the case of uniaxial tensile loading, the stress at failure is
distributed uniformly over the cross section, σ=σf, and
dA=rdldθ where l is the fiber length, r is the fiber radius and
θ is the angle as shown in Figure 7. Substituting these values
into Eq. (2) yields,
∫∫
2π l
t
F = 1 - exp -
0
σf
σo
m
rdldθ
Ao
(3)
0
where lt is the total length in tension. Integration yields the
probability of length lt failing at stress σf,
σ
m
F = 1 - exp - σ f
o
2πrlt
Ao
(4)
Thus, Eq. (4) can be used to scale the data for gauge length of
area Ao and characteristic strength σo, to lengths lt. For lt
equal to the test length, Ao = 2πrlt and Eq. (4) simplifies to
the more familiar Weibull form,
σ
m
F = 1 - exp - σ f
o
(5)
Predictions from long-length
The usefulness of this distribution is that it can be
transformed into a linear format as,
strength distributions
Optical fibers in the field experience a variety of stress
conditions over different fiber lengths. For example, in
splice enclosures, relatively short lengths of fiber are
subjected to bending; however, considering the number of
splice enclosures involved, hundreds and even thousands of
meters of fiber are under stress. In this common situation it
is important to determine which flaws pose the greatest risk
to mechanical reliability. For applications involving
kilometers of fiber under low stress, it is common to treat the
fiber as if it were no stronger than the proof stress level.
However, as fiber comes closer to the home, it is expected to
have greater mechanical reliability, thereby, requiring the
knowledge and accountability of flaw distribution at and
above the proof stress level. Since measured tensile strength
distributions typically use lengths that do not match those
deployed in service or model loading configurations such as
bending, one must scale the distribution to the length and
loading configuration appropriate for a given application.
Weibull's cumulative failure probability distribution has
found wide applicability for describing the dependence of
strength on size. The failure probability at an applied stress
is given by,7,8
F = 1 - exp -
∫
σ
σo
m
dA
Ao
(2)
where m is the Weibull slope, A is the surface area under
stress σ, and Ao is the surface area corresponding to the
characteristic strength σo.
820 International Wire & Cable Symposium Proceedings 1991
lnln 1 = m lnσf - m lnσo
1-F
(6)
where m is the Weibull slope and -mlnσo is the intercept.
Recall that data in Figure 5 are plotted in terms of lnln(1-F)
versus lnσf according to Eq. (6); however, as previously
observed, the data are not linear, but rather has a
characteristic curvature associated with a truncated
distribution.3 Theoretical models for curved distributions are
significantly more complex than Eq. (4), and the scaling of
such distributions is justified. However, here we take a more
pragmatic approach that simplifies scaling significantly,
especially in the case of bending.
The distribution in Figure 5 is not extensive enough to attain
the theoretical slope of n+1; so, for discussion purposes,
ranks 1 to 15 were fit to Eq. (6) yielding a slope of
approximately 5. Similarly, ranks 15 through 106 were fit to
Eq. (6) yielding a Weibull slope of 1.7. The composite
distribution is shown in Figure 8. The Weibull parameter σo
for each portion of the distribution is also given in Figure 8.
Using Eq. (4), where Ao is the total area for 20 meter test
lengths and 62.5 µm glass radius, the composite distribution
is scaled to a range of new lengths lt in Figure 9. Note again
that these distributions are degraded following proof testing
due to crack growth during strength testing. As expected, the
failure probability for each strength level is increased as the
in-service length increases. Also note that the shift is slightly
greater for the region of the distribution with lower m value.
Conversely, as the in-service length decreases, the probability
of encountering a flaw near the truncation strength also decreases.
However, for a typical failure probability
requirement of lx10-5, the predicted distribution shows that
even applications using only 1 meter lengths in tension need
be concerned with flaws below the 600 kpsi (4200 MPa)
Òhigh strengthÓ region shown in Figure 1. For stressed
lengths 100 kilometers and greater, Figure 9 shows that
reliability designs should be focused primarily on flaws near
the proof stress level. The data in Figure 9, when matched
with a given application, will also help focus on the type of
data needed for reliability determinations.
Bending.
The surface tensile stress due to bending of fiber also is a
reliability concern, since large stresses are easily generated.
However, bending places a considerably smaller area under
stress compared to uniaxial tension, due to the fact that only
half of the fiber surface is under tensile loading (Figure 10),
and the stress distribution over that surface is highly
nonuniform. Therefore, it is important to determine what
portion of the strength distribution is of concern for a given
bend application.
The simplest bending situation is where the entire fiber
length in bending experiences a constant bend radius. The
Weibull scaling laws for the more complex case of 2-point
bending have already been derived by Matthewson et. al.,9
and therefore, the analysis here will follow their form and
notation.
The surface tensile stresses generated by fiber bending are
dependent entirely on the bend configuration. The stress, σ,
is zero at the neutral axis, θ=0, and reaches a maximum at
θ=π/2; see Figure 7. The well known relationship between
stress, σ, and the bend radius R is given by,
σ = r sin θ E
(7)
R
where r is the fiber radius and E is its YoungÕs modulus.
YoungÕs modulus for optical glass fiber has been found to
vary linearly with strain, according to E = Eo(1+3ε), for
strength levels of concern in this study, where Eo is the zero
strain modulus.10,11 Thus, the maximum bend stress, σb, at
θ=π/2 only occurs along a thin line along the fiber lengths
and is given by,
σb = E r
(8a)
R
Therefore, the bend stress at any point on the tensile surface
is simply,
σ = σb sin θ
(8b)
The Weibull cumulative failure probability distribution for
the case of pure bending is obtained by substituting Eq. (8b)
into Eq. (2), where dA = rdldθ yielding,
∫∫
π l
b
F = 1 - exp -
0
σb sin θ
σo
m
rdldθ
Ao
(9)
Integrating over length lb reduces the above equation to,
m
σ
F = 1 - exp - σb
o
rlb
Ao
∫
π
sinmθ dθ
(10)
0
Following the derivation of Matthewson et. al.9 we let,
G(m)≡ 1
2
∫
π
0
sinmθ dθ =
∫
π
2
Γ m+1
2
sinmθ dθ = π
2 Γ m+2
2
0
(11)
where Γ(m) is the gamma function which is readily
determined using polynomial approximations.12 Substitution
of Eq. (11) into Eq. (10) gives the cumulative failure
probability distribution for bending in terms of maximum
bending stress, σb, and the length under bending, lb, for a
fiber of radius r.
σ
F = 1 - exp - σb
o
m
2rlb
G(m)
Ao
(12)
Thus, given σo and Ao from the composite 20 meter gauge
length tensile distribution in Figure 8, one can calculate the
failure probability for various lengths in bending. Figure 11
shows failure probability predictions for a range of bend
lengths.
The predictions in Figure 11 show that, similar to the tensile
distribution, the longer the bend length is, the greater the
failure probability will be. However, the predicted
distributions in bending show a lower failure probability for
the same length and stress level than that given by tensile
distributions in Figure 9. This is a consequence of the fact
that bending places fewer flaws at risk than tension. The
predicted distribution for 100 meter lengths in bending is
shown in Figure 11 to fall on top of the original 20 meter
tensile distribution. Thus, for the Weibull slopes in this data,
loading 100 meters to a constant radius in bending is
equivalent to loading 20 meters in tension. This will be
discussed further in a later section.
In light of the bend predictions we now examine a common
application that uses bending; namely, splice enclosures. The
goal here is to determine that portion of the distribution
which is most critical for making reliability predictions. As a
worse case, we assume that in a splice enclosure 1 meter of
fiber is placed in bending under a constant radius. The
predicted distribution for this application is shown in Figure
11. For a failure probability requirement of lx10-5, flaws
with strengths in the 150 kpsi (1050 MPa) range are expected
to be encountered (note that post-proof strengths are
somewhat greater than the fatigue strengths obtained in this
study). For failure probabilities < lx10-5, flaws with
strengths near the proof stress level will be encountered even
for 1 meter lengths in bending. When one accounts for all of
the fiber in splice enclosures, such a low range of failure
probabilities may not be unreasonable.
0
International Wire & Cable Symposium Proceedings 1991 821
Mean strengths in Bending and Tension.
The equivalent tensile test length, leq, for a given bend
length, lb, is determined where the mean tensile strength
equals the mean bend strength in Eq. (17),
The mean strength for a distribution of the form,
σ
F = 1 - exp - σ'
m
(13)
is given by,9
− = σ' Γ 1+ 1
σ
(14)
m
Thus for the tensile distribution in Eq. (4) the mean strength
is,
1
1
− = σ Ao m Γ
σ
1+ m
(15)
t
o
2πrlt
and similarly the mean strength for a constant bend radius
from Eq. (12) is,
−=σ
σ
b
o
Ao
2rlbG(m)
1
m
Γ 1+
1
m
(16)
The ratio of the mean strength for a fiber with a constant
bend radius to mean tensile strength is then,
−
σ
lt m1
π
b =σ
−
o
σt
G(m) lb
(17)
Thus, the mean strength in bending can be predicted given
the mean tensile strength, the ratio of tensile to bend lengths
and the Weibull slope parameter, m. This analysis is
particularly useful since mean bend strength for a constant
bend radius cannot easily be measured. Figure 12 is a plot of
the ratio of mean strengths in Eq. (17) for a 20 meter tensile
length and a range of bend lengths. As the Weibull slope, m,
increases, i.e., variability in flaw size decreases, the
difference in mean strengths decreases. For m values less
than 15 the ratio of mean strengths is a strong function of the
variability in flaw size.
An alternate expression for the failure probability in bending
is obtained by substituting the value for σo in Eq. (15) into
Eq. (12) giving,
σ
F = 1 - exp - −b
σt
m
m
1
1
Γ 1+
m
lt
π
G(m) lb
(18)
leq =
G(m)
π lb
(19)
Equation 18 is plotted in Figures 13 and 14 for the equivalent
tensile length as a function of bend length, for both long- and
short-length applications, respectively. For an m value of
approximately 15, the equivalent tensile length is nearly
1/10th the bend length. That is to say, a 1 kilometer length in
bending is equivalent to testing 100 meters in tension. This
analysis is helpful in determining the appropriate strength
testing requirements for reliability prediction. For example, a
bending application involving a total of 1000 kilometers
requires tensile data for approximately 100 kilometers of
fiber.
Summary
Long-length strength distributions are necessary for making
failure predictions at very low probabilities In this paper, a
386 kilometer strength distribution was examined in light of
theoretical distributions of proof tested fibers. This
distribution showed the beginnings of the classical truncated
distribution associated with proof testing; however, it lacked
data near the proof stress level where Weibull slopes are
believed to be on the order of n. The distribution was
extended to strength levels near the proof stress level in a
conservative fashion using conventional Weibull statistics.
This composite distribution was then scaled to a variety of
lengths for both bending and tensile applications.
Even though bending only stresses 10 to 20% of the fiber
surface when compared to tension, bending applications need
be concerned with flaws near the proof stress level,
particularly for high reliability requirements. Such analyses
are needed for reliability predictions and in determining
strength testing requirements for various tensile and bending
applications.
References
1. K.E. Lu, G.S. Glaesemann, M.T. Lee, D.R. Powers, and
J.S. Abbott, ÒMechanical and Hydrogen Characteristics of
Hermetically Coated Optical Fiber,Ó Opt. Quantum Electron.
22, 227-237 (1990).
Knowing the mean strength for a given tensile strength
distribution and the unimodal Weibull slope, one can
calculate the failure probability for a given bend stress and
bend length.
2. G.S. Glaesemann and D.J. Walter, ÒMethod of Obtaining
Long-length Strength Distributions for Reliability
Prediction,Ó Opt. Eng., 30 (6), 746-748 (1991).
Equivalent tensile length.
3. E.R. Fuller, Jr., S.M. Wiederhorn, J E. Ritter, Jr., P.B.
It is difficult to determine the flaws that are of greatest
reliability risk for bending applications, due to the
complexity of the stress distribution over the fiber surface.
Large flaws near the neutral axis are of lower risk than small
flaws at θ=π/2. It therefore is helpful to translate the bend
condition to an equivalent tensile condition.9
822 International Wire & Cable Symposium Proceedings 1991
Oates, ÒProof Testing of Ceramics, Part 2: Theory,Ó J. Mater.
Sci., 15, 2282-2295 (1980).
4. G.S. Glaesemann and S.T. Gulati, ÒDynamic Fatigue Data
for Fatigue Resistant Fiber in Tension vs. Bending,Ó in 1989
Technical Digest Series. Vol. 5, Proc Optical Fiber
Communication Conference, 48 (1989).
5. G.S. Glaesemann, M.L. Elder, and J.W. Adams,
ÒDependence of fatigue resistance on Titania doping of
silica-clad optical fiber,Ó presented at Optical Fiber
Communication Conference, post-dead paper PD 13-1 PD13-4 (1990) .
Figure 2.
If Every 20 m Specimen Stressed to Only 400 kpsi (2800 MPa)
Test Development Approach:
Avoid testing flaws that do not pose a threat to mechanical reliability
Strength (MPa)
6. J.E. Ritter, Jr, N. Bandyopadhyay, and K. Jakus,
ÒStatistical Reproducibility of the Dynamic and Static
Fatigue Experiments,Ó Amer. Ceram. Soc. Bul. 60 (8) 798 806 (1981).
70
210
350
700
1400 2100 2800 3500 4900 7000
0.999
20 m Gauge Length
Ambient Environment
0.95
Represents 16.5 km Data
0.5
0.2
7. W. Weibull, ÒStatistical Distribution Function of Wide
Applicability,Ó J. Appl. Mech., 18(3) 293-297 (1951).
Failure
Probability
0.05
0.02
8. W.D. Scott and A. Gaddipati, ÒWeibull Parameters and the
Strength of Long Glass Fibers,Ó in Fracture Mechanics of
Ceramics, Vol. 3, edited by R.C. Bradt, D.P.H. Hasselman
and F.F. Lange (Plenum Press, New York,1978) pp. 125-42.
0.005
Failed
Passed
0.002
0.0005
10
30
50
100
200
300 400 500
700 1000
Strength (kpsi)
9. M.J. Matthewson, C.R. Kurkjian, and S.T. Gulati,
ÒStrength Measurement of Optical Fibers by Bending,Ó J.
Am. Ceram. Soc., 69 (11) 815-821 (1986).
10. G. S. Glaesemanm, S. T. Gulati, and J. D. Helfinstine,
ÒThe Effect of Strain and Surface Composition on YoungÕs
Modulus of Optical FibersÓ, Optical Fiber Communication
Conference, 1988 Technical Digest Series, Vol.1 (Optical
Society of America, Washington, DC 1988), p.26.
11. J.T. Krause, L.R. Testardi, and R.N. Thurston,
ÒDeviations from Linearity in the Dependence of Elongation
upon Force for Fibers of Simple Glass Formers and of Glass
Optical Lightguides,Ó Phys. Chem. Glasses 20(6) 135-139
(1979)
Figure 3.
Schematic of Continuous Fiber Strength Test Apparatus
x
x
x
Payout
Bx
For
Normal
Proof
Test
Operation
x
x
x
Pulley
Load Cell
Take-Up
Neumatic
Slide
Fiber
Proof Test Machine
12. M. Abramowitz and I.A. Stegun, Handbook of
Mathematical Functions. Dover, New York, 1964.
Figure 1.
Strength Distribution of Standard Silica-Clad Fiber
A
Tension Stand
with Load Cell
Figure 4.
Loading Cycle of 20 Meter Gauge Length
Strength (MPa)
70
210
350
700
Passed
1400 2100 2800 3500 4900 7000
0.999
Represents 16.5 km Data
0.5
Failure
Load
0.2
Failure
Probability
Failure During Loading
20 m Gauge Length
Ambient Environment
0.95
0.05
Load
1986
0.02
0.005
Time
Time
0.002
0.0005
10
30
50
100
200
300 400 500
700 1000
Strength (kpsi)
International Wire & Cable Symposium Proceedings 1991 823
Figure 8.
Composite Long-Length Strength Distribution
Figure 5.
Strength Distribution of 386 Kilometers of
Titania-Doped Silica-Clad Fiber
Strength (MPa)
Strength (MPa)
140
0.999
0.9
0.5
0.1
350
210
700
1400
2800 4200
140
0.999
0.9
0.5
7000
20 m Gauge Length
Ambient Environment
350
1400
700
2800 4200
7000
m = 4.9
σo = 520 kpsi
0.1
0.01
0.01
Failure
Probability
0.001
Failure
Probability
210
0.0001
0.001
m = 1.7
σo = 6650 kpsi
0.0001
0.00001
0.00001
0.000001
0.000001
20 m Gauge Length
Ambient Environment
1e-07
1e-07
1e-08
1e-08
20
30
50
100
200
400
600
20
1000
30
50
100
200
400
600
1000
Strength (kpsi)
Strength (kpsi)
Figure 6.
Theoretical Strength Distribution after Proof Testing
Figure 9.
Failure Probability Predictions for Tension
Reference 3.
Strength (MPa)
140
0.999
0.9
0.5
Slope = m
Ln Ln
1
1-F
Failure
Probability
Initial Strength
Distribution
210
350
0.1
100 km
0.01
10 km
0.001
1 km
0.0001
100 m
0.00001
20 m
700
1400
2800 4200
7000
Slope = n - 2
0.000001
1m
20 m Gauge Length
Ambient Environment
1e-07
σfmin
1e-08
20
30
50
100
400
200
600
1000
Strength (kpsi)
Ln σf
Figure 10.
Fiber in Bending
Figure 7.
Cross-Section of Fiber
Tension
y
r sin θ
Neutral
Axis
θ
r
824 International Wire & Cable Symposium Proceedings 1991
r
M
M
Compression
Figure 14.
Equivalent Tensile Test Length for Bent Fiber
Figure 11.
Failure Probability Predictions for Bending
Short-Lengths
Strength (MPa)
210
140
0.999
1400
700
2800 4200
7000
180
0.5
160
0.1
140
100 km
0.01
120
Equivalent
Tensile
Length (m)
10 km
0.001
Failure
Probability
350
1 km
0.0001
20 m tension
100 m
0.00001
Weibull
Modulus, m
leq/lb
m=5
m = 15
m = 30
m = 45
0.17
0.10
0.07
0.06
100
80
60
40
0.000001
20 m Gauge Length
Ambient Environment
1m
1e-07
20
0
1e-08
20
30
50
100
200
400
600
1000
0
200
400
600
800
1000
Bend Length (m)
Strength (kpsi)
Figure 12.
Predicted Mean Bend Strength
20 Meter Tensile Measurement
4.0
Bend Length
3.5
2m
20 m
200 m
1 km
10 km
3.0
Mean
Bend Strength
2.5
2.0
Mean
Tensile Strength
1.5
1.0
0.5
G. Scott Glaesemann, SP-DV-01-8 Corning Inc., Corning,
N.Y., 14831. Glaesemann is a senior development
engineer responsible for the optical waveguide strength
laboratory at Corning. He has been employed by Corning
for five years at the Sullivan Park research and
development facility. Glaesemann received his masterÕs
degree and Ph. D. in mechanical engineering from the
University of Massachusetts and a B. S. in mechanical
engineering from North Dakota State University. He is a
member of the American Ceramic Society.
0
0
10
5
15
20
25
30
35
40
45
50
Weibull Modulus, m
Figure 13.
Equivalent Tensile Test Length for Bent Fiber
Long-Lengths
9000
Weibull
Modulus, m
leq/lb
m=5
m = 15
m = 30
m = 45
0.17
0.10
0.07
0.06
8000
7000
6000
Equivalent
Tensile
Length (m)
5000
4000
3000
2000
1000
0
0
5
10
15
20
25
30
35
40
45
50
Bend Length (km)
International Wire & Cable Symposium Proceedings 1991 825